Hello, I'm doing a free online finance course at coursera.
The latest assignment has a question with the following:
price - 81,110
face value - 100,000
YTM - 11.5%
semi annual coupon for 8 years.
How much is the coupon?
The instructor showed how to calculate it with Excel but I'd like to know what the math involved is.
Thanks for any help,
Gerard
The price is what is called Present Value (PV). The face value and the semi annual coupons you get are what is called Future Values (FV). To turn a PV into a FV you accumulate the interest for the lenght of time wanted, that is
FV = PV (1+i)
n
where i is the interest rate per period and n is the number of periods. You can turn this around also
PV = FV / (1+i)
n
So, let's see what buying the bond will get you: It will get you interest every 6 months at the coupon rate for 8 years or you will get 16 FV payments p for an amount of 100000*(i/2). To turn each of the payments p into a PV at the current interest rate of 11.5% per annum we divide each p by 1.0575
n where n is the time until you get you payment, i.e. n= 1, 2, 3, ... 16. The total sum of thaose payments are
PV
1 = p (1/1.0575 + 1/1.0575
2 +1/1.0575
3 + ... + 1/1.0575
16) = p (1 - 1/1.0575
16)/0.575
which is a standard financial formula. Note also that at the end of 8 years you will get a lump sum payment of $100000 whose PV at the current interest rate of 11.5% per annum is
PV
2 = 100000/1.0575
16
The total present value PV
1 + PV
2 = $81,100 the present price of the bond. Thus you have
$81,100 = 100000/1.0575
16 + p (1 - 1/1.0575
16)/0.0575
Solve for p = 100000*(i/2) and then for i.